In this blog post, we will look at how our understanding of reality has changed through the differences between classical mechanics and quantum mechanics.
According to classical mechanics, if the initial state of motion of an object is known precisely, regardless of its size, the state of the object after a certain time can be measured accurately, and two mutually exclusive states cannot coexist. However, quantum mechanics, which emerged in the 20th century, revealed that mutually exclusive states can coexist in the microscopic world.
To understand the coexistence of mutually exclusive states in the microscopic world, consider a spinning top with a radius of 5 cm in the macroscopic world. The spinning top will be spinning in either a clockwise or counterclockwise direction. The direction of rotation of the spinning top is already determined before observation, and it is only known through observation. In contrast, imagine a spinning top as small as an electron in the microscopic world. The direction of rotation of this spinning top can be either clockwise or counterclockwise. The two states coexisting in a single spinning top are determined by observation to be one direction of rotation. Which of the two directions will be determined cannot be known before observation. Unlike the macroscopic world, in the microscopic world governed by quantum mechanics, mutually exclusive states coexist before we observe them. Einstein found it difficult to accept the concept that the coexistence of mutually exclusive states and observation itself determines the state of an object, and he was skeptical of the interpretation of quantum mechanics, saying, “Does the moon not exist before you see it?”
This characteristic of quantum mechanics has not only stimulated theoretical curiosity, but has also had a significant impact on technological development. Recently, research is being conducted on quantum computers that perform ultra-fast calculations by applying the coexistence of mutually exclusive states. This is a good example of how the coexistence of mutually exclusive states in quantum mechanics can be realized in reality. These research results on the microscopic world fundamentally question the common sense ideas we have naturally acquired about the macroscopic world. For example, the principle of quantum computers works in a completely different way from that of conventional classical computers, opening up amazing possibilities that go beyond our everyday understanding. Similar questions can be found in logic.
Classical logic is a logical system with only two truth values: “true” and “false.” In classical logic, any statement is either “true” or “false.” This fits well with our common sense. However, according to Priest, there are statements that are “true” and “false” at the same time, in addition to statements that are “true” or “false.” To explain this, he presents the “liar’s paradox.” To understand the liar’s paradox, let’s distinguish between self-referential statements and non-self-referential statements. A self-referential statement is, as the name suggests, a statement that refers to itself. For example, the true statement “This sentence consists of eighteen syllables” refers to itself and states how many syllables it consists of. On the other hand, the true statement “The capital of Peru is Lima” only states where the capital of Peru is and does not refer to itself.
“This sentence is false” is a liar sentence. This is a self-referential sentence in which the expression ‘this sentence’ refers to the sentence itself and says that it is ‘false.’ Then why does Priest think that liar sentences should be considered ‘true and false at the same time’? To answer this question, let’s first assume that liar sentences are ”true.” In that case, the liar’s sentence is “false.” This is because the liar’s sentence refers to itself and says that it is “false.” On the other hand, let’s assume that the liar’s sentence is “false.” In that case, the liar’s sentence is “true.” This is because that is what the sentence says. According to Priest, in any case, the liar’s statement is a statement that is “true and false at the same time.” Therefore, he believes that the liar’s statement must be given the value “true and false at the same time.” He presents various examples that support the existence of truth values that are “true and false at the same time” other than the liar’s statement. In particular, he believes that the coexistence of mutually exclusive states in quantum mechanics suggests this point.
Since classical logic cannot handle sentences with truth values that are “true and false at the same time,” Priest proposed LP*, one of the non-classical logics that can handle such sentences. However, in LP, some intuitively appealing inference rules do not hold. Consider the example of the affirming the antecedent rule. In classical logic, the affirming the antecedent rule holds. This means that if the conditional statement “P is Q” is true, then its antecedent P must also be true, and its consequent Q must also be true. Similarly, in LP, for the rule of affirming the antecedent to hold, the conditional statement and its antecedent P must both be true or both false, and its consequent Q must also be true or both true and false. However, in LP, if the antecedent of the conditional statement is “true and false” and the consequent is “false,” then both the conditional statement and the antecedent are “true and false,” but the consequent is “false.” Although the antecedent affirmation rule does not hold, LP is significant as an attempt to answer fundamental questions about classical logic.
These developments in quantum mechanics and logic have provided an important opportunity to rethink our existing knowledge system. The characteristics of the microscopic world presented by quantum mechanics are not only of interest to scientists, but also have an important influence on philosophical and logical discussions. Through this, we will be able to gain a deeper understanding of nature and the universe, and lead future technological innovations.
